General superpositions of Gaussian beams and propagation errors

Abstract:

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [Math. Comp. 82 (2013), pp. 919–952] can be applied to obtain the propagation error of order $k^{1- \frac {N}{2}- \frac {d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.